Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals,their uncertainties,and correlations. Part I: Theory

Computation of phase diagrams in mineral systems and quantitative geothermobarometry thrive on the availability and accuracy of internally consistent thermodynamic datasets for minerals. The prevailing two methodologies applied to derive them, mathematical programming (MAP) and least squares regression (REG), have their very specific advantages and deficiencies which are to some extent complementary. Bayes estimation (BE), the novel technique proposed here for obtaining internally consistent thermodynamic databases, can combine the advantages of both MAP and REG but avoid their drawbacks. It optimally uses the information on thermochemical, thermophysical, and volumetric properties of phases and experimental reaction reverals to refine the thermodynamic data and returns their uncertainties and correlations. Therefore, BE emerges as the method of choice. The theoretical background of BE, and its relation to MAP and REG, is explained. Although BE is conceptually simple, it can be computationally demanding. Fortunately, modern computer technology and new stochastic methods such as Gibbs sampling help surmount those difficulties. The basic ideas behind these methods are explored and recommendations for their use are made using the Al₂SiO₅ unary as an example. The potential of BE and its future perspective for application to multicomponent-multiphase systems appear very promising. For the convenience of readers not interested in the mathematical details of BE, an illustrative example is given in the Appendix to promote an intuitive understanding of what BE is all about.     

Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals,their uncertainties,and correlations. Part II: Application

Internally consistent thermodynamic data, including their uncertainties and correlations, are reported for 22 phases of the quaternary system CaO-Al₂O₃-SiO₂-H₂O. These data have been derived by simultaneous evaluation of the appropriate phase properties (PP) and reaction properties (RP) by the novel technique of Bayes estimation (BE). The thermodynamic model used and the theory of BE was expounded in Part I of this paper. Part II is the follow-up study illustrating an application of BE. The input for BE comprised, among others, the a priori values for standard enthalpy of formation of the i-th phase, Δ_f H _i ⁰ , and its standard entropy, S _i ⁰ , in addition to the reaction reversal constraints for 33 equilibria involving the relevant phases. A total of 269 RP restrictions have been processed, of which 107 turned out to be non-redundant. The refined values for Δ_f H _i ⁰ and S _i ⁰ obtained by BE, including their 2σ-uncertainties, appear in Table 4; the Appendix reproduces the corresponding correlation matrix. These data permit generation of computed phase diagrams with 2σ-uncertainty envelopes based on conventional error propagation; Fig. 3 depicts such a phase diagram for the system CaO-Al₂O₃-SiO₂. It shows that the refined dataset is capable of yielding phase diagrams with uncertainty envelopes narrow enough to be geologically useful. The results in Table 4 demonstrate that the uncertainties of the prior values for Δ_f H _i ^Emphasis>0 , given in Table 1, have decreased by up to an order of magnitude, while those for S _i ⁰ improved by a factor of up to two. For comparison, Table 4 also lists the refined Δ_f H _i ⁰ and S _i ⁰ data obtained by mathematical programming (MAP), minimizing a quadratic objective function used earlier by Berman (1988). Examples of calculated phase diagrams are given to demonstrate the advantages of BE for deriving internally consistent thermodynamic data. Although P-T curves generated from both MAP and BE databases will pass through the reversal restrictions, BE datasets appear to be better suited for extrapolations beyond the P-T range explored experimentally and for predicting equilibria not constrained by reversals.     

An internally consistent thermodynamic dataset with uncertainties and correlations: 2. Data and results

Abstract This, the second of two papers, represents the application of a least squares approach, discussed in the previous paper, to the generation of an internally consistent thermodynamic dataset involving 60 reactions among 43 phases, in the system K₂O–Na₂O–CaO–MgO–Al₂O₃–SiO₂–H₂O–CO₂. We make the assumption that all the thermodynamic data, with the exception of enthalpies of formation of the phases, are well known, and solve for an internally consistent set of enthalpies which reproduces the 60, experimentally determined, phase equilibrium reactions. An important difference between our dataset and that of previous alternatives in the literature is that we are able to determine the uncertainties on, and correlations between, the enthalpies of formation for all phases in the set, and hence are able to apply simple error propagation techniques to determine the uncertainties in any phase equilibrium calculations performed using this dataset. Selection of reactions, for geothermometry and geobarometry, may be more readily made by choosing equilibria with small uncertainties in their thermodynamics. Our data are in reasonably close agreement with the high temperature molten oxide calorimetry results on silicate minerals where available, a fact which lends a degree of confidence to the results.     

An internally consistent thermodynamic dataset with uncertainties and correlations: 1. Methods and a worked example

Abstract This, the first two papers, sets out the philosophy and methods of determining an internally consistent thermodynamic dataset for minerals using the least squares method. The applicability of the least squares method is discussed, and it is applied to a small set of experimental equilibria in the system Na₂O–Al₂O₃–SiO₂–H₂O. The importance is stressed of defining not only the enthalpies of formation of minerals, but also the uncertainties and the correlations among them. The system which has been used as an illustration for this paper serves as a visual guide to the method, as it is small enough to represent graphically in two dimensions. In the paper which follows, we extend the method to a system of 60 equations (experimentally determined equilibria) involving 34 unknowns (enthalpies of formation of mineral end-members).     

Importance of considerations of mixing properties in establishing an internally consistent thermodynamic database: thermochemistry of minerals in the system Mg2SiO4-Fe2SiO4-SiO2

A thermodynamic solution model is developed for minerals whose compositions lie in the two binary systems Mg₂SiO₄-Fe₂SiO₄ and Mg₂Si₂O₆-Fe₂Si₂O₆. The formulation makes explicit provision for nonconvergent ordering of Fe²⁺ and Mg²⁺ between M1 and M2 sites in orthopyroxenes and non-zero Gibbs energies of reciprocal ordering reactions in both olivine and orthopyroxene. The calibration is consistent with (1) constraints provided by available experimental and natural data on the Fe-Mg exchange reaction between olivine and orthopyroxene ± quartz, (2) site occupancy data on orthopyroxenes including both crystallographic refinements and Mössbauer spectroscopy, (3) enthalpy of solution data on olivines and orthopyroxenes and enthalpy of disordering data on orthopyroxene, (4) available data on the temperature and ordering dependence of the excess volume of orthopyroxene solid solutions, and (5) direct activity-composition determinations of orthopyroxene and olivine solid solutions at elevated temperatures. Our analysis suggests that the entropies of the exchange [Mg(M2)Fe(M1)Fe(M2)Mg(M1)] and reciprocal ordering reactions [Mg(M2)Mg(M1)+ Fe(M2)Fe(M1)Fe(M2)Mg(M1)+Mg(M2)Fe(M1)] cannot differ significantly (± 1 cal/K) from zero over the temperature range of calibration (400°–1300° C). Consideration of the mixing properties of olivine-orthopyroxene solid solutions places tight constraints on the standard state thermodynamic quantities describing Fe-Mg exchange reactions involving olivine, orthopyroxene, pyralspite garnets, aluminate spinels, ferrite spinels and biotite. These constraints are entirely consistent with the standard state properties for the phases-quartz,-quartz, orthoenstatite, clinoenstatite, protoenstatite, fayalite, ferrosilite and forsterite which were deduced by Berman (1988) from an independent analysis of phase equilibria and calorimetric data. In conjunction with these standard state properties, the solution model presented in this paper provides a means of evaluating an internally consistent set of Gibbs energies of mineral solid solutions in the system Mg₂SiO₄-Fe₂SiO₄-SiO₂ over the temperature range 0–1300° C and pressure interval 0.001–50 kbars. As a consequence of our analysis, we find that the excess Gibbs energies associated with mixing of Fe and Mg in (Fe, Mg)₂SiO₄ olivines, (Fe, Mg)₃Al₂Si₃O₁₂ garnets, (Fe, Mg)Al₂O₄ and (Fe, Mg)Fe₂O₄ spinels, and K(Mg, Fe)₃AlSi₃O₁₀(OH)₂ biotites may be satisfactory described, on a macroscopic basis, with symmetric regular solution type parameters having values of 4.86±0.12 (olivine), 3.85±0.09 (garnet), 1.96±0.13 (spinel), and 3.21±0.29 kcals/gfw (biotite). Applications of the proposed solution model demonstrate the sensitivity of petrologic modeling to activity-composition relations of olivine-orthopyroxene solutions. We explore the consequences of estimating the activity of silica in melts forming in the mantle and we develop a graphical geothermometer/geobarometer for metamorphic assemblages of olivine+orthopyroxene+quartz. Quantitative evaluation of these results suggests that accurate and realistic estimates of silica activity in melts derived from mantle source regions,P-T paths of metamorphism and other intensive variables of petrologic interest await further refinements involving the addition of trace elements (Al³⁺ and Fe³⁺) to the thermodynamic formulation for orthopyroxenes.     

Extension of lattice strain theory to mineral/mineral rare-earth element partitioning: An approach for assessing disequilibrium and developing internally consistent partition coefficients between olivine, orthopyroxene, clinopyroxene and basaltic melt

Olivine/melt and orthopyroxene/melt rare-earth element (REE) partition coefficients consistent with clinopyroxene/melt partition coefficients were determined indirectly from subsolidus partitioning between olivine, orthopyroxene, and clinopyroxene after suitable correction for temperature. Heavy- and middle-REE ratios for olivine/clinopyroxene and orthopyroxene/clinopyroxene pairs correlate negatively with effective cationic radius, whereas those for the light REEs correlate positively with cationic radius, generating a U-shaped pattern in apparent mineral/clinopyroxene partition coefficients versus cationic radius. Lattice strain models of partitioning modified for subsolidus conditions yield negative correlations of olivine/clinopyroxene and orthopyroxene/clinopyroxene with respect to cationic radii, predicting well the measured partitioning behaviors of the heavy and middle REEs but not that of the light REEs. The light-REE systematics cannot be explained with lattice strain theory and, instead, can be explained by disequilibrium enrichment of the light REEs in melt inclusions or on the rims of olivine and orthopyroxene. Realistic light-REE partition coefficients were thus extrapolated from the measured heavy- and middle-REE partition coefficients using the lattice strain model. Light REE olivine/melt and orthopyroxene/melt partition coefficients calculated in this manner are lower than most published values, but agree reasonably well with partitioning experiments using the most recent in situ analytical techniques (secondary-ionization mass spectrometry and laser ablation inductively coupled plasma mass spectrometry). These new olivine/melt and orthopyroxene/melt partition coefficients are useful for accurate modeling of the REE contents of clinopyroxene-poor to -free lithologies, such as harzburgitic residues of melting. Finally, the application of the lattice strain theory to subsolidus conditions represents a framework for assessing the degree of REE disequilibrium in a rock.     

An improved and extended internally consistent thermodynamic dataset for phases of petrological interest,involving a new equation of state for solids

The thermodynamic properties of 254 end‐members, including 210 mineral end‐members, 18 silicate liquid end‐members and 26 aqueous fluid species are presented in a revised and updated internally consistent thermodynamic data set. The PVT properties of the data set phases are now based on a modified Tait equation of state (EOS) for the solids and the Pitzer & Sterner (1995) equation for gaseous components. Thermal expansion and compressibility are linked within the modified Tait EOS (TEOS) by a thermal pressure formulation using an Einstein temperature to model the temperature dependence of both the thermal expansion and bulk modulus in a consistent way. The new EOS has led to improved fitting of the phase equilibrium experiments. Many new end‐members have been added, including several deep mantle phases and, for the first time, sulphur‐bearing minerals. Silicate liquid end‐members are in good agreement with both phase equilibrium experiments and measured heat of melting. The new dataset considerably enhances the capabilities for thermodynamic calculation on rocks, melts and aqueous fluids under crustal to deep mantle conditions. Implementations are already available in thermocalc to take advantage of the new data set and its methodologies, as illustrated by example calculations on sapphirine‐bearing equilibria, sulphur‐bearing equilibria and calculations to 300 kbar and 2000 °C to extend to lower mantle conditions.     

The partial molar volume, thermal expansivity, and compressibility of H2O in NaAlSi3O8 liquid: new measurements and an internally consistent model

The molar volumes of 19 hydrous albitic liquids (1.9 to 6.1 wt% H₂O^total) were determined at one bar and 505–765 K. These volume data were derived from density measurements on hydrous glasses at 298 K, followed by measurements of the thermal expansion of each glass from 298 K to its respective glass transition temperature. The technique exploits the fact that the volume of a glass is equal to that of the corresponding liquid at the limiting fictive temperature (T _f′), and that T _f′ can be approximated as the temperature near the onset of the rapid increase in thermal expansion that occurs in the glass transition interval. The volume data of this study were combined with available volume data for anhydrous, Na₂O-Al₂O₃-SiO₂ liquids to derive the partial molar volume (±1) of the H₂O component in an albitic melt at ∼565 K and one bar. To extend the determination of to higher temperatures and pressures, the molar volumes of the hydrous albitic liquids determined in this study were combined with those measured by previous authors at 1023–1223 K and 480–840 MPa, leading to the following fitted values (±1) at 1673 K and one bar: (±0.46)×10⁻³ cm⁻³/mol-K, and dVˉ ^H ₂ ^{O total} /dP=−3.82 (±0.36)×10⁻⁴ cm³/mol-bar. The measured molar volumes of this study and those of previous authors can be recovered with a standard deviation of 0.5%, which is within the respective experimental errors. There is a significant difference between the values for derived in this study as a function of temperature and pressure and those obtained from an existing polynomial, primarily caused by the previous absence of accurate density measurements on anhydrous silicate liquids. The coefficients of thermal expansion (=4.72×10⁻⁴/K) and isothermal compressibility ( _T =1.66×10⁻⁵/bar) for the H₂O component at 1273 K and 100 MPa, indicate that H₂O^total is the single most expansive and compressible component in silicate liquids. For example, at 1473 K and 70 MPa (conditions of a mid-ocean ridge crustal magma chamber), the presence of just 0.4 wt% H₂O will decrease the density of a basaltic liquid by more than one percent. An equivalent decrease in melt density could be achieved by increasing the temperature by 175 degrees or the decreasing pressure by 230 MPa. Therefore, even minor quantities of dissolved water will have a marked effect on the dynamic properties of silicate liquids in the crustal environment. Received: 20 August 1996 / Accepted: 15 March 1997     

The SiO2 polymorphs: The equations of state and thermodynamic properties of phase transformations

Thermal equations of state have been derived for polymorphic forms of SiO₂ and values of \(\mathop \smallint \limits_0^P\) V _T dP have been tabulated. Available experimental data on the phase equilibria at high pressures and temperatures have been used with these equations to calculate the standard thermodynamic functions for the α-quartz-coesite and coesite-stishovite transformations. A study of sensitivity of calculated thermodynamical properties to uncertainties of phase equilibria data, initial data for elastic constants and equations of state has been carried out. The discrepancies between standard thermodynamic properties of these transformations calculated from phase equilibria data and solution calorimetry data still persist.     

Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures

A revised regular solution-type thermodynamic model for twelve-component silicate liquids in the system SiO₂-TiO₂-Al₂O₃-Fe₂O₃-Cr₂O₃-FeO-MgO-CaO-Na₂O-K₂O-P₂O₅-H₂O is calibrated. The model is referenced to previously published standard state thermodynamic properties and is derived from a set of internally consistent thermodynamic models for solid solutions of the igneous rock forming minerals, including: (Mg,Fe²⁺,Ca)-olivines, (Na,Mg,Fe²⁺,Ca)^M2 (Mg,Fe²⁺, Ti, Fe³⁺, Al)^M1 (Fe³⁺, Al,Si)₂ ^TETO₆-pyroxenes, (Na,Ca,K)-feldspars, (Mg,Fe²⁺) (Fe³⁺, Al, Cr)₂O₄-(Mg,Fe²⁺)₂ TiO₄ spinels and (Fe²⁺, Mg, Mn²⁺)TiO₃-Fe₂O₃ rhombohedral oxides. The calibration utilizes over 2,500 experimentally determined compositions of silicate liquids coexisting at known temperatures, pressures and oxygen fugacities with apatite ±feldspar ±leucite ±olivine ±pyroxene ±quartz ±rhombohedral oxides ±spinel ±whitlockite ±water. The model is applicable to natural magmatic compositions (both hydrous and anhydrous), ranging from potash ankaratrites to rhyolites, over the temperature (T) range 900°–1700°C and pressures (P) up to 4 GPa. The model is implemented as a software package (MELTS) which may be used to simulate igneous processes such as (1) equilibrium or fractional crystallization, (2) isothermal, isenthalpic or isochoric assimilation, and (3) degassing of volatiles. Phase equilibria are predicted using the MELTS package by specifying bulk composition of the system and either (1) T and P, (2) enthalpy (H) and P, (3) entropy (S) and P, or (4) T and volume (V). Phase relations in systems open to oxygen are determined by directly specifying the f _{o 2} or the T-P-f _{o 2} (or equivalently H-P-f _{o 2}, S-P-f _{o 2}, T-V-f _{o 2}) evolution path. Calculations are performed by constrained minimization of the appropriate thermodynamic potential. Compositions and proportions of solids and liquids in the equilibrium assemblage are computed.     

A statistical-chemical and thermodynamic approach to the study of lunar mineralogy

Principal components analysis is used to study the chemical compositions of pyroxenes of five Apollo 12 specimens. Important correlations recognized in the variation of oxide weight per cent are: MGO, Al₂O₃, SiO₂| CaO, TiO₂, FeO MgO, Al₂O₃, SiO₂| FeO MgO, SiO₂, FeO | Al₂O₃, CaO, TiO₂ where the oxides on one side of the bar are correlated positively with each other and negatively with the oxides on the other side. Several other similarly distinct relationships with significantly less variance could be noted. These correlations indicating substitutional relationships can be interpreted as representative of stable and metastable trends of crystallization by using crystal-chemical and thermodynamic information. The per cent variance of pyroxene groups with characteristic trends in each specimen can be evaluated and interpreted in terms of history of crystallization. Distribution of Fe and Mg in certain pairs of olivine and pyroxene, which are found in contact in the rock and which may have crystallized simultaneously, is useful in recognizing the tendency towards chemical equilibrium in FeMg distribution during a limited interval in the liquidus or subsolidus stages.     

Phlogopite: High temperature solution calorimetry,thermodynamic properties,Al-Si and stacking disorder,and phase equilibria

Methods have been developed for solution calorimetry of hydrous phases in molten lead borate near 700°C. These involve thermochemical cycles using dissolution and decomposition reactions of hydrous silicates and hydroxides. Preliminary results suggest that H₂O derived from the decomposition of hydroxides dissolves in molten 2PbO-B₂O₃ with an exothermic enthalpy of solution of −5.7 ±0.7 kcal mol⁻¹. Hydroxyphologopite persists metastably at 714°C and its heat of solution in 2PbO·B₂O₃ has been measured. From these new data, the standard enthalpy of formation of phlogopite from the elements at 25°C is −1485.5 ±1.5 kcal mol⁻¹. The standard free energy of formation is -1394.6 ±1.5 kcal mol⁻¹, assuming complete tetrahedral Al-Si disorder.Two structural features complicate the thermodynamics of synthetic and natural micas. The first is a varying degree of tetrahedral Al-Si disorder. Raman spectroscopic study of phlogopite synthesized above 600°C suggests a disordered Al-Si distribution. Calculations of the P-T locus of the geologically important equilibrium: Phl + 3Qtz = 3En + Sa + H₂O, using our thermochemical data, agree within experimental error with the results of calculations based on the best available phase equilibrium data only if a tetrahedrally disordered phlogopite is assumed. Such calculations are very sensitive to uncertainties in ΔH° and ΔG°, and reversed phase equilibrium experiments remain essential to obtaining reliable estimates of thermodynamic properties. In contrast to these Al-Si disordered phlogopites, some biotites of low temperature parageneses (<600°C) may have substantial Al-Si order. A variable Al-Si distribution has a substantial effect on the configurational entropy and therefore on the free energy of the mica in question. Because of these and other questions, applications of biotite equilibria to determining volatile fugacities in igneous and metamorphic petrogenesis are subject to large uncertainties.The second structural complication is stacking disorder, which is present in phlogopite synthesized at 650°C but not in the 850°C sample. The enthalpy difference between these two samples, determined by solution calorimetry, is smaller than the experimental uncertainty of ±1.0 kcal mol⁻¹. Thus there appears to be little driving force for ordering, and micas with disordered stacking sequences may persist in many geologic environments. The effect of stacking disorder on thermodynamic properties is probably very small.     

A thermodynamic theory of the Grüneisen ratio at extreme conditions: MgO as an example

The Grüneisen ratio, γ, is defined as γy=αK _TV/C_v. The volume dependence of γ(V) is solved for a wide range in temperature. The volume dependence of αK _T is solved from the identity (? ln(αK _T)/? ln V)T ≡ δ _T-K′. α is the thermal expansivity; K _T is the bulk modulus; C _V is specific heat; and δ _Tand K′ are dimensionless thermoelastic constants. The approach is to find values of δ _T and K′, each as functions of T and V. We also solve for q=(? ln γ/? ln V) where q=δ _T -K′+ 1-(? ln C _V/? ln V)T. Calculations are taken down to a compression of 0.6, thus covering all possible values pertaining to the earth's mantle, q=? ln γ/? ln V; δ _T=? ln α/? ln V; and K′= (?K _T/?P)T. New experimental information related to the volume dependence of δ _T, q, K′ and C _V was used. For MgO, as the compression, η=V/V ₀, drops from 1.0 to 0.7 at 2000 K, the results show that q drops from 1.2 to about 0.8; δ _T drops from 5.0 to 3.2; δ _T becomes slightly less than K′; ? ln C _V/? In V→0; and γ drops from 1.5 to about 1. These observations are all in accord with recent laboratory data, seismic observations, and theoretical results.     

Experimental determination of the reaction paragonite=jadeite+kyanite+H2O,and internally consistent thermodynamic data for part of the system Na2O−Al2O3−SiO2−H2O,with applications to eclogites and blueschists

Hydrothermal reversal experiments have been performed on the upper pressure stability of paragonite in the temperature range 550–740 ° C. The reaction $$\begin{gathered} {\text{NaAl}}_{\text{3}} {\text{Si}}_{\text{3}} {\text{O}}_{{\text{1 0}}} ({\text{OH)}}_{\text{2}} \hfill \\ {\text{ paragonite}} \hfill \\ {\text{ = NaAlSi}}_{\text{2}} {\text{O}}_{\text{6}} + {\text{Al}}_{\text{2}} {\text{SiO}}_{\text{5}} + {\text{H}}_{\text{2}} {\text{O}} \hfill \\ {\text{ jadeite kyanite vapour}} \hfill \\ \end{gathered}$$ has been bracketed at 550 ° C, 600 ° C, 650 ° C, and 700 ° C, at pressures 24–26 kb, 24–25.5 kb, 24–25 kb, and 23–24.5 kb respectively. The reaction has a shallow negative slope (? 10 bar °C^?1) and is of geobarometric significance to the stability of the eclogite assemblage, omphacite+kyanite. The experimental brackets are thermodynamically consistent with the lower pressure reversals of Chatterjee (1970, 1972), and a set of thermodynamic data is presented which satisfies all the reversal brackets for six reactions in the system Na₂O-Al₂O₃-SiO₂-H₂O. The Modified Redlich Kwong equation for H₂O (Holloway, 1977) predicts fugacities which are too high to satisfy the reversals of this study. The P-T stabilities of important eclogite and blueschist assemblages involving omphacite, kyanite, lawsonite, Jadeite, albite, chloritoid, and almandine with paragonite have been calculated using thermodynamic data derived from this study.     

The contribution of A.E. Kontorovich to the theory of petroleum generation

Aleksei Emil'evich Kontorovich, a titular member of the Russian Academy of Sciences, contributes very much to oil geology, in particular, to the foundation and development of the modern theory of petroleum generation and methodology of petroleum potential forecast.     

Bearthite,Ca2Al(PO4)2OH: stability,thermodynamic properties and phase relations

Contributions to Mineralogy and Petrology - The new phosphate bearthite, Ca2Al(PO4)2HO, found in high-pressure metamorphic rocks, has been synthesized from a stoichiometric mixture of γ-Al2O3...     

Preliminary phase relations involving glaucophane and applications to high pressure petrology: new heat capacity and thermodynamic data

New heat capacity measurements and cell volume data are presented for a very magnesian glaucophane from a Tauern Window eclogite. These data are combined with estimated entropy, thermal expansion, and compressibility data to generate an enthalpy of formation for glaucophane from experimentally determined phase equilibria. The data are supported by preliminary experiments of the author and provide consistent calculations on the pressure of formation of the Tauern eclogites and on the position of the blueschist-greenschist transformation reaction as studied experimentally by Maruyama et al. (1986). The resulting thermodynamic data for glaucophane may be combined with the dataset of Holland and Powell (1985) to calculate phase relations for blueschists and eclogites. The stability of magnesian glaucophane lies in the pressure range between 8 and 32 kbars at 400° C and between 13 and 33 kbars at 600° C, and the unusual eclogite assemblage of glaucophane+kyanite from the Tauern Window is restricted to pressures above 20 kbars at high water activity.